§1Everything from p
A standard thread has only two independent dimensions — its major diameter and its pitch. Every other number on the drawing is generated from them by fixed geometry.
That is the quiet miracle behind interchangeability. Write M10×1.5 and you have said everything: the 60° form is fixed by the standard, the fundamental triangle’s height follows from the pitch, and the pitch diameter, both minor diameters, the thread depth, the stress area and the tap drill all fall out as d minus a constant times p. The constants — 0.6495, 1.0825, 1.2269, 0.9382 — look arbitrary on a data sheet and are nothing of the kind: each is a simple fraction of the triangle height H, and §2 derives every one. The practical payoff is that a machinist, a designer or an inspector can reconstruct any 60° thread’s working dimensions from the designation alone, with four multiplications — no tables, no lookup, no trust in a figure someone else copied. This page is that reconstruction, worked end to end on the series’ house thread, the M10×1.5, and it underwrites the two pages either side of it: the same arithmetic ran the Unified page’s inch numbers, and the measuring page ahead will need d2 as its target.
Contents§2Where the constants come from
Draw the sharp 60° triangle of height H = 0.866 p and truncate it at the standard fractions — the constants are those fractions, doubled onto a diameter.
Work down the hero’s ladder, remembering that a depth per side counts twice on a diameter. The pitch line sits ⅜H below the sharp crest by construction of the equal-width point, and the crest itself is cut back H/8 to the major diameter — so from major to pitch is ¼H + ⅛H = ⅜H per side… taken together, (d − d2)/2 = ⅜H, giving d2 = d − 0.75 H = d − 0.6495 p (0.75 × 0.86603 = 0.6495 exactly). The internal minor D1 lies where the nut’s thread stops, ⅝H of engagement below the major on each side: D1 = d − 1.25 H = d − 1.0825 p. The external minor d3 lies deeper still, at the rounded root of the bolt: d3 = d − 1.2269 p, about 0.708H per side. And the sharp apex itself is never cut — the last fraction of the vee exists only as construction, exactly like the hero’s dashed lines. Every one of these is checkable in one line of arithmetic, which is the standard this page holds itself to: 0.75 × 0.86603 = 0.6495; 1.25 × 0.86603 = 1.0825; and the average of the pitch and minor constants, (0.6495 + 1.2269)/2 = 0.9382, is a number §4 is about to need.
Contents§3An M10×1.5, computed in full
Apply the constants to the series’ standing example and the whole thread appears from two numbers.
| Quantity | Formula | Value |
|---|---|---|
| Triangle height | H = 0.86603 p | 1.299 mm |
| Pitch diameter | d2 = d − 0.6495 p | 9.026 mm |
| Internal minor | D1 = d − 1.0825 p | 8.376 mm |
| External minor | d3 = d − 1.2269 p | 8.160 mm |
| Stress area | As = (π/4)(d − 0.9382 p)² | 58.0 mm² |
| Tap drill | ≈ d − p | 8.5 mm → 77.0% |
| Notice the ordering the geometry enforces: d2 above both minors, and the nut’s minor D1 above the bolt’s d3 — the bolt’s root dives below where the nut’s thread reaches, which is exactly the clearance that lets the rounded, fatigue-friendly root of §2 exist without interference. These six numbers are the thread; everything a drawing, a tap chart or a wire measurement will ever say about an M10×1.5 is one of them, toleranced. | ||
§4The stress area
A bolt does not break at its minor diameter — the threads reinforce the core a little — so strength is computed on an effective diameter midway between pitch and minor.
For the M10×1.5: (d2 + d3)/2 = (9.026 + 8.160)/2 = 8.593 mm, and As = (π/4) × 8.593² = 58.0 mm² — precisely the number the metric fasteners page multiplied by property-class stresses to get the 37.1 kN yield and 46.4 kN ultimate loads of an M10 8.8. That closes the loop between the sections: the fasteners pages took As on faith as a catalogue figure, and this page shows it is nothing but the 60° triangle again — the average of two ladder rungs, squared. The physical reading is worth keeping too: tested to destruction, a threaded bar breaks as if it were a plain bar of diameter d − 0.9382 p, a little fatter than its true minor section, because the thread helix bridges and stiffens the core. Strength lives between the rungs.
§5Tap drills and the 77% rule
The hole drilled before tapping sets how much thread the nut member actually gets — and the universal workshop rule, drill = d − p, is a deliberate 77%, not an approximation of 100%.
Drill an M10×1.5 hole at the rule’s 8.5 mm (d − p) and the engagement is (10 − 8.5)/(1.299 × 1.5) = 77.0% — the rule is the 77% figure, exactly, for every metric pitch. Drill 8.4 mm instead and engagement rises to 82.1%. Why not chase 100%? Because the last quarter of engagement buys almost nothing and costs almost everything: beyond about three-quarters depth, the §8-section stripping calculations barely improve — the nut fails by thread shear at nearly the same load — while tapping torque climbs steeply as the tap is asked to cut into the shrinking root space, and in tough materials the small drill is how taps snap off in holes. Around 75% engagement is therefore the deliberate standard for general work; soft materials that strip easily justify drilling smaller for 80–85%, and tough or work-hardening ones justify 65–70% for the tap’s survival. The tapping page ahead turns this dial into practice; here the point is that the dial exists, and that d − p is its calibrated 77% mark.
§6Fine pitch arithmetic
Every formula on this page has p in it, so shrinking the pitch moves every dimension at once — and the direction is always toward more core, shallower helix and finer control.
Rework the M10 at the fine pitch of 1.25 mm and the ladder compresses: every depth is a constant times a smaller p, so d2, D1 and d3 all rise toward the major, and the stress area grows to As = (π/4)(10 − 0.9382 × 1.25)² = 61.2 mm² — 5.5% more than the coarse thread’s 58.0, from the identical bar of steel, the metric twin of the Unified page’s +12.7% at half an inch (the inch gap is wider because 13→20 TPI is a bigger pitch jump than 1.5→1.25). The lead angle falls in proportion, tightening the systems page’s self-locking margin; the thread depth falls, so a fine thread lives happily in a thin wall where a coarse one would break through; and the tap drill moves up (d − p = 8.75 mm), leaving more parent metal around the hole. The costs are the mirror image — shallow threads that tolerate less damage and dirt, slower assembly, and worse behaviour in soft parents — which is why the arithmetic of this page never chooses the pitch by itself: it prices both options, and the duty decides.
Contents§7Quick reference
The working core of the page on one card rack.
The triangle
H = 0.866 p
constants = fractions of H
The ladder
d2 = d − 0.6495 p
D1 − 1.0825 · d3 − 1.2269
M10×1.5
9.026 · 8.376 · 8.160 mm
As = 58.0 mm²
Tap drill
d − p → exactly 77.0%
8.4 mm → 82.1%
Fine pitch
M10×1.25 → 61.2 mm²
+5.5% core, shallower helix